On the other hand, due to the SVD truncation step, conservation properties are loss. Several techniques exist in the literature to correct conservation errors for low rank methods. In [1]}, the low rank solution is rescaled so that the total mass is conserved, and a similar mass correction technique is proposed in [2]} for a dynamical low rank method.
In [3]}, moment fitting is applied to the low rank solution so that the corrected moments match those solved from the macroscopic fluid equations.
In [4]}, a dynamical low rank method with Lagrangian multipliers is developed to improve conservation properties for the total mass and momentum as well as local projected moment equations. More recently, along the same line, the truly local conservation of mass, momentum, and energy is attained for the dynamical low rank method [5]}. The idea is to fix certain basis functions in the dynamical low rank approximation and employ a modified Petrov–Galerkin formulation which is compatible with the remainder of the approximation.
In our recent work [6]}, a conservative SVD truncation is developed via an orthogonal projection to a subspace with conservation of macroscopic moments followed by a weighted SVD truncation performed on the remainder term. As a result, local mass and momentum conservation is achieved at the discrete level. However, the algorithm does not enjoy global or local energy conservation, as the associated full rank scheme can not conserve energy. In fact, an implicit symplectic time discretization is usually needed for exact energy conservation of a fully discrete scheme [7]}.
